Question

Solve for x and y; if x>o and y>0:
log xy= lof x/y +2 log 2 = 2

Answer 1

log xy= log x/y +2 log 2 = 2

taking the first two;
log xy= lof x/y +2 log 2
⇒ log x + log y = log x - log y + 2 log 2
⇒ log y = -log y + 2 log 2
⇒ 2 log y = 2 log 2
⇒ y = 2

taking first and third;
log xy = 2
⇒ log 2x = 2
⇒ 2x = 10² = 100
⇒ x = 100/2 = 50

x = 50, y = 2

Answer 2

Let us say that the base of the logarithm is 10.

Log x/y + 2 Log 2 = 2 =  2 * Log 10
     Log x/y = 2 Log 10 - 2 Log 2  = Log 10² - Log 2² = Log 10²/2² = Log 25
    x/y = 25    => x = 25 y

Log xy = 2      =>    Log 25y² = 2 
     =>  Log (5y)² = 2     =>  2 Log (5y) = 2     
     Log 5y =1 
    5 y = 10     => y = 2
      x = 50

verification:

Log xy = Log 100  =2  ;  log x/y + 2 Log 2  = Log 25 + Log 4 = Log 100 = 2